Problem V. To find the diameter and circumference of a circle, either from the other. RULE 1. As 7 is to 22, so is the diameter to the circumference, and as 22 is to 7, so is the circumference to the diameter. RULE 2. As 113 is to 355, so is the diameter to the circumference, and as 355 is to 13, so is the circumference to the diameter. RULE 3. As I is to 3.1416, so is the diameter to the circumference, and as 3.14.6 is to 1, so is the circumference to the diameter. RULE.--Multiply half the circumference by half the diameter, and the product will be the area. *These three methods do not exactly agree, but the last is the most correct. The exact proportion between the diameter and circumference of a circle has not yet been ascertained. + The following are some of the most useful problems relating to the circle 3. Square of circumference x .07958 area. 5. As 88: 7 :: square of circumference: area. 6. Diameter .8862-side of an equal square. .2251➡side of an inscribed square. 1.128 diameter of an equal circle. u bos The area of a circle given to find the diameter and circumference. RULE.—1. Divide the area by .7854, and the square root of the quotient will be the diameter. 2. Divide the area by .07958, and the square root of the quotient will be the circumference. Problem VIII. To find the area of an oval, or ellipsis. RULE.-Multiply the longest and shortest diameters together, and the product by .7854; the last product will be the area.* *The longest diameter of an ellipsis is called the transverse, and the shortest the conjugate diameter. Examples. 1. What is the area of an oval whose longest diameter is 5 feet, and shortest 4 feet? 5X4X.78545.708 ft. Ans. 2. What is the area of an oval whose longest diameter is 21, and shortest 17 ? Problem IX. To find the area of a globe or sphere. Ans. 280.3878. RULE.-Multiply the circumference by the diameter, and the product will be the area. Examples. 1. How many square feet in 3. What is the area of the surthe surface of a globe whose di- | face of a cannon shot, whose diameter is 14 inches and circum-ameter is one inch? ference 44? 44 x 14-616 Aus. 2. How many square miles in the earth's surface, its circumference being 25000, and its diameter 7957 miles? Ans. 198943750. Ans. 3.1416 inches. 4. How many square inches in the surface of an 18 inch artificial globe? Ans. 1017.8784. 2. Fensuration of Solids. 1. A solid is a figure having three dimensions, viz. length, breadth and thickness. 2. A prism is a body whose ends are any equal and similar plane figures, and whose sides are parallelograins. 3. A cube is a body having six equal sides, all of which are squares. 4. A parallelopipedon is a body having six rectangular sides, ev ery opposite pair of which are equal and parallel. 5. A cylinder is a round prism, having circles for its ends. 6. A pyramid is a solid whose base is any plane figure, and whose sides are triangular, meeting in a point at the top called a vertex. 7. A cone is a round pyramid, having a circle for its base. 8. A sphere is a solid bounded by one continued convex surface, every part of which is equally distant from a point within called the centre. Mensuration of Solids teaches to determine the spaces included by contiguous surfaces, and the sum of the measures of these including surfaces is the whole surface of the body. The measure of a solid is called its solidity, capacity, or content. The content is estimated by the number of cubes, whose sides are inches, or feet, or yards, &c. contained in the body. Problem I. To find the solidity of a cube. RULE-Cube one of its sides, that is, multiply the side by itself, and that product by the side again, and the last product will be the RULE.-Multiply the length by the breadth, and that product by the depth, the last product will be the answer. Examples. 1. What is the content of a parallelopipedon whose length is 6 feet, its breadth 24 feet, and its depth 13 feet? 6x2.5 x 1.75=26.25 or 264 feet. 2. How many feet in a stick of hewn timber 30 feet long, 9 inches broad, and 6 inches thick? Ans. 114 feet. Problem III. To find the side of the largest square stick of timber that can be hewn from a round log. RULE.-Extract the square root of twice the square of the semidiameter at the smallest end for the side of the stick when squared⚫ To find the solidity of a prism, or cylinder. RULE.-Multiply the area of the end by the length of the prism, for the content. Examples. 1. What is the content of a triangular prism, the area of whose end is 2.7 feet, and whose length is 12 feet? 2.7 X 12 32.4 ft. Ans. 2. What number of cubick feet in a round stick of timber, whose diameter is 18 inches, and length 20 feet? Ans. 35.343. Problem V. To find the solidity of a pyramid or cone. RULE.-Multiply the area of the base by the height, and one third of the product will be the content. Examples. 1. What is the content of a cone whose height is 124 feet, and the diameter of the base 24 feet? 21×2×25-64 and 61.7854 × 124÷÷÷3—20.453125. 2. What is the content of a triangular pyramid, its height being 14 feet, and the sides of its base being 5, 6, and 7 feet? Ans. 71.035+ Problem VI. To find the solidity of a sphere.* RULE.-Multiply the cube of the diameter by .5236, or multiply the square of the diameter by the circumference. Examples. 1. What is the content of a sphere whose diameter is 12 in 2. What is the solid content of the earth, its circumference ches? 12 x 12 x 12 x 5236 being 25000 miles? 904.7808 Ans. Ans. 263858149120 miles. 3. Gauging. Gauging teaches to measure all kinds of vessels, as pipes, hogs-1 heads, barrels, &c. RULE. To the square of the bung diameter add the square of the head diameter; multiply the sum by the length, and the product by .0014 for ale gallons, or by .0017 for wine gallons. *The surface of a sphere is found by multiplying its diameter by its circum ference. |